### Venn Diagram Symbols

In mathematics, a set is a collection of things. Sets are often denoted using the curly brackets {}, for example the set of the first 5 odd numbers is {1, 3, 5, 7, 9}. There are two definitions that you should know up front. If we have two sets, A and B, then

- Acup B is the union of A and B. It is the collection of all the things that are contained in A or B (or both).
- Acap B is the intersection of A and B. It is the collection of things that are contained in both A and B.

In a more informal way, cup means “or” and cap means “and”. Understanding sets is essential for Venn diagrams.

### Venn Diagrams

Now we know more about our set notation, we can move on to how to display sets using Venn diagrams.

A typical Venn diagram looks like the picture on the left. One circle represents the set A, whilst the other represents B. The section where the two circles cross over is for all things that are contained in both sets, i.e. it is Acap B.

The box is called the universal set and is denoted by the Greek letter xi, and it contains all of the objects we’re concerned with, including objects that aren’t contained within neither A nor B. Such things belong in the section that is inside the rectangle but outside the circle.

## Example 1: Understanding Sets

Let A={1, 5, 8, 10} and B={2, 3, 5, 7, 8}. Work out which numbers are in Acup B and then in Acap B.

Firstly, Acup B is the collection of all numbers that are in either of them (or both of them). In other words, all we need to do is combine the two sets, ignoring any repeats. So, we get

Acup B={1, 2, 3, 5, 7, 8, 10}

Now, for Acap B we want all numbers that are in both sets. Looking, we can see that there are only two: 5 and 8. Therefore,

Acap B={5, 8}.

## Example 2: Drawing Venn Diagrams

We have universal set xi={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} Let A be the set of odd numbers and let B be the set of prime numbers. Draw a Venn diagram for this information.

Firstly, we must establish sets and . Which of the numbers from the universal set are odd? Answer: 1, 3, 5, 7, 9, and 11. Which of the numbers from the universal set are prime? Answer: 2, 3, 5, 7, and 11. There’s some crossover here – 3, 5, 7, and 11 are contained in both sets, which means they go in the part where the circles intersect.

What about the numbers that are odd but aren’t prime? There are two: 1 and 9. These will go in the part of circle A that doesn’t intersect with B. Similarly, there is 1 number that is prime but isn’t odd: 2. It will go in the part of circle B that doesn’t intersect with A.

Finally, all the numbers that are neither odd nor prime: 4, 6, 8, and 10, will go inside the rectangle but outside either circle. The result is the Venn diagram on the right.

## Example 3: Using Venn Diagrams

The Venn diagram below shows A is a set of odd numbers and B is a set of prime numbers. Use the Venn diagram to determine the following:

a) The probability of a number that is selected at random being both odd and prime?

b) The probability of selecting an odd number that isn’t a prime number?

a) Looking at the Venn diagram, we see that 4 of the 11 numbers are in both A and B, so the probability of a number being both odd and prime is frac{4}{11}.

b) We are looking for all numbers that are in set A but not set B, 1 and 9, which is frac{2}{11}.

## Example 4: Venn Diagrams and Conditional Probability

The Venn diagram below shows G represents people selecting Geography and B is people selecting History. Use the Venn diagram to determine the following:

a) Find Gcup H

b) Find Htext{’}

c) text{P}(G|H)

part a) So, Gcup H means “G or H”, meaning we’re interested firstly in all people to take Geography or History (or both). The total number is 6+12+5=23, and therefore,

text{P}(Gcup H)=dfrac{23}{32}

Part b) We need Htext{’} which means all people who don’t take History. From the diagram we can see that 6+9=15 people don’t take it, so we get that

text{P}(Htext{’})=dfrac{15}{32}

Part c) It is only for higher students and is a question of something called conditional probability. The | line means “given”, so text{P}(G|H) means “the probability of G given H”. In other words, the probability that someone takes Geography given that we already know they take History. Venn diagrams are useful for these questions. If we know someone takes History, then they must be one of the 5+12=17 people. Then, there are 12 out of those 17 people who take Geography, or in other words

text{P}(G|H)=dfrac{12}{17}

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